Theoretical And Applied Study Of Symplectic Multi-Resolution Time Domain Scheme

In recent years, with the rapid development of the computer performance and the emergence of various new algorithm in computational mathematics and computational physics, computational electromagnetics have shown a situation of unprecedented prosperity.The traditional FDTD method has been widely applied to broad-band, transient, and full-wave analyses owing to its simplicity, generality, and facility for parallel computing.Due to the constraints of the numerical dispersion and the numerical stability,the method is needed to consume much memory and computing time in the calculation of electrically large size of the electromagnetic problems.As a result, the FDTD method is restricted the computation of the electrically large problems.Many efforts have been made in relaxing or removing the above two constraints in order to reduce the computational expenditures. Among them,the multiresolution time-domain (MRTD) method,proposed in1996,has achieved low numerical dispersion with numerical grid resolutions as low as two points per wavelength. However,the CFL stability condition still remains. In order to further explore efficient methods for optimum electromagnetic simulation, other improved time strategies are proposed. For example, the high-order Runge-Kutta (R-K) approach was introduced in. However, the approach is dissipative and needs large amount of memory. Another alternative method is the alternating direction implicit (ADI) algorithm. Although it saves CPU time owing to unconditional stability, undesirable numerical precision and dispersion will happen once the high CFL number is adopted.Along with the people to the understanding of the physical nature of the problem, to realize in the pursuit of the algorithm with high precision, also should strive to maintain the original system intrinsic properties.Due to the electromagnetic field equations can be transformed into an infinite dimensional Hamilton system, Hamilton system has the intrinsic properties of a series of, so in the numerical solution of the Hamilton system, it is particularly important to maintain its intrinsic properties.Symplectic method is a new numerical method to maintain the intrinsic nature of Hamilton system, the algorithm of numerical calculation in long time, has a computational advantage there is nothing comparable to this common numerical methods.In this paper, the symplectic algorithm is introduced to the electromagnetic calculations and combined with higher-order spatial characteristics of traditional MRTD.As a result, symplectic Multi-Resolution Time Domain scheme,a high efficient computational electromagnetic method.is developed.This dissertation systematically studies the symplectic Multi-Resolution Time Domain scheme. For temporal direction, the high-order symplectic integration scheme is used to keep the global symplectic structure of Maxwell’s equations for long-term simulation. For spatial direction, the equations were evaluate with multi-resolution approximations to reduce numerical dispersion and improve numerical precision. For grid generation, the multi-region decomposition technology and the concept of effective dielectric constant are proposed to model material discontinuities. With the help of these matched schemes and techniques, we can build a fast, low-consumed, and accurate time-domain solver.Focusing on the "theoretical and applied study of symplectic Multi-Resolution Time Domain scheme", The main researches and contributions are made as follows:(1) The MRTD schemes based on Daubechies scaling functions" are studied theoretically.The iterative equations of the electromagnetic fields are derived in detail.(2) The symplectiness of Maxwell’s equations in free space is discussed. It is verified that the time evolution matrix of Maxwell’s equations is symplectic matrix and conserves the total energy of electromagnetic field. Then, the symplectic algorithm is combined with higher-order spatial characteristics of traditional MRTD.As a result, symplectic Multi-Resolution Time Domain scheme,a high efficient computational electromagnetic method,is developed. The iterative equations of the electromagnetic fields are derived in detail.(3) Numerical dispersion and stability are compared for a variety of high-order time-domain schemes. In particular, through numerical experiments on long-term simulation, energy conservation, and numerical precision, the advantages of the symplectic Multi-Resolution Time Domain scheme are demonstrated.(4) The key technology of symplectic multi-resolution time-domain algorithm in electromagnetic simulation is investigated,which includes:is introduction of plane wave source, absorbing boundary conditions, the near to far field transformation technique,etc.(5) According to the multi-region decomposition technology and the concept of effective dielectric constant,a conformal symplectic multi-resolution time-domain scheme based on Daubechies scaling functions used to dielectric targets is proposed.The numerical results of the electromagnetic scattering computation of dielectric targets show that out scheme can solve the discontinuous surface in dielectric case and the staircase error of Yee’s leapfrog meshing,and also can improve computational efficiency and accuracy obviously.